Lecture 7: ARIMA Model Process

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Lecture 7 ARIMA Model Process

• The following topics will be covered
• Properties of Stock Returns
• AR model
• MA model
• ARMA
• Non-Stationary Process
• Seasonal Models
• Regression Models with Time Series Errors

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Time Series Plot
data msi set crsp.msi mretvwretd
mno(year(date)-1930)12month(date) keep mno
mret proc gplot datamsi symbol vnone ijoin
l1 plot mretmno run
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Basics

• Return measures
• Empirical properties of returns (page 16 Table
  1.2 and Figure 1.4)
• Daily returns of market indexes and individual
  stocks tend to have high excess kurtoses
• The standard deviation of monthly returns is
  greater than the standard deviation of daily
  returns
• The difference between simple and log returns is
  not substantial.

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Stationarity

• A time series rt is said to be strictly
  stationary if the joint distribution rt1, ,
  rtk is identical to that of rt1t, , rtkt
  for all t, where k is an arbitrary positive
  integer.
• See page 9 to 10 of RT for the definition of
  joint distribution
• A time series rt is said to be weakly
  stationary if both the mean of rt and the
  covariance between rt and rt-l are
  time-invariant, where is an arbitrary integer.
  That is (1) E(rt)µ
• (2) Var(rt) ?0
• (3) cov(rt,
  rt-l)?l

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Correlation and Auto-correlations
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Portmanteau Statistic
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Alternative Time Series
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Properties of AR Models

• Mean and Variance of AR(1)
• See page 30 derive them
• Autocorrelation Function an AR(1) model
• Mean, Variance, and ACF of AR(2) page 29
• Stationarity of AR(2)
• AR(p) model

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AR(2)
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Identifying AR Models
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Estimation of AR(p)

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Forecasting AR(p)
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More on Forecasting AR(p)

• 2) 2-step Ahead Forecast
• See page 40 the variance of the forecast error
  is Vareh(2) Model Checking If the model is
  adequate, then the residual series should behave
  as a white noise. The ACF and the Portmanteau
  test of the residual at can be used to check
  the closeness of at to a white noise.
• proc arima datamsi identify varmret nlag12
• estimate p1 methodml
• forecast lead1
• run

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Moving-Average (MA) Models
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ARMA Models
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Random Walk and ARIMA

• Random Walk without a Drift ptpt-1at
• Random Walk with a drift ptµpt-1at
• ARIMA page 60

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Seasonal Models

• Seasonality of quarterly earnings
• Seasonal adjustments yt-yt-s(1-Bs)yt
• Multiplicative Seasonal Models

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Regressions with Time Series Errors
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Exercises

• (1) Ch 2, 6
• (2) Get quarterly earnings per share (data11)
  from Quarterly Computstat file. Do the following
• (a) Compute average EPS across all firms
• (b) Plot average quarterly EPS from 1996 through
  2004
• (c) Plot ACF and PACF for quarterly EPS
• (d) Take seasonal difference and check ACF and
  PACF again